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Chapter 6: Problem 68

According to the American Veterinary Medical Association, \(30 \%\) of Americans own a cat. a. Find the probability that exactly 2 out of 8 randomly selected Americans own a cat. b. In a random sample of 8 Americans, find the probability that more than 3 own a cat.

Short Answer

Expert verified
a. The probability that exactly 2 out of 8 randomly selected Americans own a cat is approximately 0.296. b. The probability that more than 3 out of 8 randomly selected Americans own a cat is approximately 0.350.

Step by step solution

01

Calculate the probability that exactly 2 out of 8 randomly selected Americans own a cat.

To find this probability, use the binomial probability formula \(P(2; 8, 0.3) = C(8, 2) \cdot (0.3^2) \cdot ((1 - 0.3) ^ {8 - 2})\). Calculate each part separately: \[C(8, 2) = \frac{8!}{2!(8 - 2)!} = 28\], \[(0.3^2) = 0.09, \] \[((1 - 0.3) ^ 6) = 0.017]\. Then, plug these values into the binomial probability formula to get the probability that exactly 2 out of 8 randomly selected Americans own a cat.
02

Calculate the probability that more than 3 out of 8 randomly selected Americans own a cat.

To find this probability, you need to subtract the probability that 3 or fewer Americans own a cat from 1. This is equivalent to calculating \(1 - P(0; 8, 0.3) - P(1; 8, 0.3) - P(2; 8, 0.3) - P(3; 8, 0.3)\). For each calculation, use the binomial probability formula as shown in Step 1. Afterwards, sum up the results of these four calculations and subtract the sum from 1.

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